Asymptotic lifting for completely positive maps

• Published in: Journal of functional analysis Vol. 287; no. 12; p. 110655
• Main Authors: Forough, Marzieh, Gardella, Eusebio, Thomsen, Klaus
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.12.2024
• Subjects: C-algebras; Completely positive map; Group action; Lift; Matematik; Mathematics; Group action; Completely positive map; C-algebras; Lift
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2024.110655

Let A and B be C⁎-algebras with A separable, let I be an ideal in B, and let ψ:A→B/I be a completely positive contractive linear map. We show that there is a continuous family Θt:A→B, for t∈[1,∞), of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then Θt can be chosen to have this property asymptotically. If A and B carry continuous actions of a second countable locally compact group G such that I is G-invariant and ψ is equivariant, we show that the family Θt can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that Θt is linear and completely positive for all t∈[1,∞). In the equivariant setting, if A, B and ψ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if G is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.